In this work, approximations to the solutions of singularly perturbed second-order linear delay differential equations are studied. We firstly use two-term Taylor series expansion for the delayed convection term and obtain a singularly perturbed ordinary differential equation (ODE). Later, an efficient and simple asymptotic method so called Successive Complementary Expansion Method (SCEM) is employed to obtain a uniformly valid approximation to this corresponding singularly perturbed ODE. As the final step, we employ a numerical procedure to solve the resulting equations that come from SCEM procedure. In order to show efficiency of this numerical-asymptotic hybrid method, we compare the results with exact solutions if possible; if not we compare with the results that are obtained by other reported methods.

This paper is concerned with two-point boundary value problems for singularly perturbed nonlinear ordinary differential equations. The case when the solution only has one boundary layer is examined. An efficient method so called Successive Complementary Expansion Method (SCEM) is used to obtain uniformly valid approximations to this kind of solutions. Four test problems are considered to check the efficiency and accuracy of the proposed method. The numerical results are found in good agreement with exact and existing solutions in literature. The results confirm that SCEM has a superiority over other existing methods in terms of easy-applicability and effectiveness.

A simple and efficient method that is called Successive Complementary Expansion Method (SCEM) is applied for approximation to an unstable two-point boundary value problem which is known as Troesch’s problem. In this approach, Troesch’s problem is considered as a singular perturbation problem. We convert the hyperbolic-type nonlinearity into a polynomial-type nonlinearity using an appropriate transformation, and then we use a basic zoom transformation for the boundary layer and finally obtain a nonlinear ordinary differential equation that contains SCEM complementary approximation. We see that SCEM gives highly accurate approximations to the solution of Troesch’s problem for various parameter values. Moreover, the results are compared with Adomian Decomposition Method (ADM) and Homotopy Perturbation Method (HPM) by using tables.

We demonstrate that the streamline-upwind/Petrov-Galerkin (SUPG) formulation enhanced with YZ𝛽 discontinuity-capturing, that is, the SUPG-YZ𝛽 formulation, is an efficient and robust method for computing 2D shallow-water equations (SWEs). The SUPG-stabilized semi-discrete formulation is discretized in time by employing the backward Euler time-integration scheme. The nonlinear equation systems arising from the space and time discretizations are handled using the Newton-Raphson (N-R) method at each time step. The resulting linear equation systems are solved directly at each nonlinear iteration. Two challenging test problems are provided to examine the performance of the proposed formulation and techniques. To that end, we consider a full dam-break and a partial dam-break problem. We develop the solvers in the FEniCS environment. Test computations reveal that the SUPG-YZ𝛽 formulation successfully eliminates spurious oscillations that cannot be captured with the SUPG-stabilized formulation alone in narrow regions where steep gradients occur. INTRODUCTIONMany real-world phenomena, including water flows in open channels, the transport of chemical species, flood flows (including those caused by tsunami waves, tidal flows, and storm surges), dam-break problems, and turbulence phenomena in the atmosphere and oceans can be mathematically modeled using the shallow-water equations (SWEs). Finite element methods (FEMs) are well-suited for simulating such models since they typically have complex flow domains. The flexible nature of the FEM is essential for real-world computations since we might need to work on extremely complicated geometries, for example, tidal waves or tsunamis that could happen in a zigzagging bay. A good number of studies based on various FEMs, as well as finite difference methods (FDM) and finite volume methods (FVMs), have been reported in the literature for computing SWEs. Although it is not possible to mention all of them, the following paragraphs attempt to provide a comprehensive review of featured research for computing SWEs with a particular emphasis on FEMs. For more details, the interested reader can refer to the materials provided in these studies.The authors of Refs. [1, 2] employed the standard Galerkin FEM (GFEM) with a two-step time-integration scheme they called the "selective lumping scheme" for computing SWEs. For computing two-layer SWEs, Hua and Thomasset [3] proposed a semi-implicit time-integration scheme paired with triangular elements, where the velocity components are approximated with nonconforming linear elements, and the (water) elevation is approximated with linear conforming elements. In Nessyahu and Tadmor [4], the authors proposed a second-order central difference scheme for solving nonlinear systems of hyperbolic conservation laws and evaluated the performance of the proposed formulation on Riemann-type problems. Bermudez et al. used a Lagrange-Galerkin FEM to upwind the convection term in Bermudez et al. [5]. They performed their computations on a 1D test problem and provid...